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IRLF 


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LIBRARY 

OF  THE 

UNIVERSITY  OF  CALIFORNIA 


GIFT  OF 


A  METHOD  OF  PETROGRAPHIC  ANALYSIS 


BASED  UPON 


Chromatic  Interference  with  Thin 

Sections  of  Doubly -Refracting  Crystals 

in  Parallel  Polarized  Light. 


Presented  to  the  Faculty  of  Philosophy  of  the  University  of  Penn- 
sylvania in  partial  fulfillment  of  the  requirements  for 
the  degree  of  Doctor  of  Philosophy. 


BY 
HOMER  MUNRO  DERR,  A.  M, 


The  Randal  Morgan  Laboratory  of  Physics. 
1903. 


Plate  1 


155477 


PLATE  1. 
Thin  Section  showing  Free  Gold  in  Metamorphic  Diorite 

FROM   ENCAMPMENT,    WYOMING. 

Magnified  50  diameters  :  — /,  feldspar,   ( oligoclase,  albite )  ;    q,  quartz 

h,  hornblende  ;  ra,  magnetite ;   g,  gold. 
Crossed  nicols  ;  thickness  of  section,  56  microns. 


Plate  I. 


BEHAVIOR  OF  ISOMETRIC  CRYSTALS. 

Isometric  crystals  on  account  of  their  isotropic  character  ex- 
hibit no  special  phenomena  in  polarized  light.  If  a  thin  plate 
of  such  a  mineral  be  placed  in  the  path  of  a  polarized  beam  of 
light  between  the  polarizer  and  the  analyzer,  the  beam  will  ex- 
perience no  alteration  of  its  plane  of  vibration,  no  matter  in 
what  direction  the  plate  was  cut  from  the  crystal,  nor  in  what 
position  it  lies  between  the  polarizer  and  the  analyzer  ;  for  the 
elasticity  of  the  ether  is  the  same  in  all  directions  in  such  a  case, 
and  rotating  the  mineral  about  any  axis  whatever  will  effect  no 
change  of  the  plane  of  vibration  of  the  polarized  light.  If  the 
section  is  colorless,  it  will  not  alter  the  color  or  the  brightness 
of  the  field  of  view  in  a  polarization  microscope,  except  for  the 
small  amount  of  absorption  which  a  ray  of  light  will  experience 
in  passing  through  any  medium.  If  the  section  is  colored,  the 
field  of  view  will  appear  slightly  different  in  color  from  that  of 
the  mineral ;  however,  this  color  does  not  change  by  altering 
the  position  of  the  plate  in  any  way.  Furthermore,  when  the 
nicols  are  crossed,  a  section  on  the  stage  of  the  microscope 
appears  dark,  and  revolving  it  in  any  plane  produces  no  effect. 
It  appears  light,  and  experiences  no  change  in  any  position  be- 
tween parallel  nicols.  Sections  of  transparent  isometric  crys- 
tals may  always  be  recognized  as  such  by  the  fact  that  they 
behave  as  an  amorphous  substance  in  polarized  light.  There 
are,  however,  some  optical  anomalies  in  this  connection.  Sec- 
tions of  amorphous  or  isometric  minerals  sometimes  appear 
light  when  viewed  between  crossed  nicols.  These  results  are 
due  to  internal  strains  caused  either  by  inclusions  of  gases  or 
fluids  which  exert  a  pressure  on  their  surroundings,  or  by  con- 
traction of  adjacent  parts.  These  phenomena  may  usually  be 
distinguished  from  ordinary  double  refraction,  since  the  appear- 
ance is  not  generally  uniform  throughout  the  section. 

INTERFERENCE. 

Fresnel  and  Arago  investigated  completely  the  conditions  of 
interference  of  two  rays  polarized  at  right  angles  to  each  other 


after  they  had  been  brought  back  to  the  same  plane  of  polari- 
zation by  passing  them  through  a  crystal  of  calc-spar  whose 
principal  section  made  an  angle  of  45°  with  the  planes  of  polari- 
zation of  each  of  the  two  rays.  They  found,  (1)  that  two  rays 
polarized  at  right  angles  to  each  other,  which  have  come  from 
an  unpolarized  ray,  do  not  interfere  even  when  they  are 
brought  into  the  same  plane  of  polarization  ;  (2)  that  two  rays 
polarized  at  right  angles,  which  have  come  from  a  polarized 
ray,  interfere  when  they  are  brought  back  to  the  same  plane  of 
polarization. 

Let  a  ray  of  polarized  monochromatic  light  be  incident  at  the 
lower   surface   of  a  plane-parallel  doubly-refracting  plate  at 

any  angle,  Fig.  1,  at  the 
point  A.  It  is  separated 
into  two  rays  AD  and  AE, 
vibrating  in  planes  at  right 
angles  to  each  other  and 
following  different  paths  in 
the  plate.  On  emergence 
they  follow  paths  parallel 
to  their  direction  at  inci- 
dence, but  are  not  coinci- 
dent, and  do  not  produce 
interference.  But  other 
parallel  rays  from  the  same 

-„       1  source  and  incident  at  B 

and  C,  will  emerge  from 

the  upper  surface,  so  that  from  all  points  E  and  F,  the 
ordinary  component  of  one  ray  and  the  extraordinary  of  an- 
other follow  the  same  path.  These  rays  will  have  travelled 
over  slightly  different  paths  in  the  plate,  and  with  different 
velocities ;  then  on  emergence  one  ray  must  have  advanced 
a  certain  number  of  wave-lengths,  or  fractions  thereof,  ahead 
of  the  other.  The  waves  are  therefore  in  different  phases  and 
retain  this  difference  of  phase  while  they  travel  in  the  same 
medium.  The  plane  of  vibration  of  the  extraordinary  ray  is 
parallel  to  the  principal  optic  section  of  the  plate,  and  that  of 
the  ordinary  ray  is  at  right  angles  to  the  same.  If  an  analyzer 
be  placed  in  the  path  of  these  rays,  each  ray  will  be  by  it 
resolved  into  components  the  vibrations  of  which  are  in  and  at 
right  angles  to  the  principal  section  of  the  polarizer,  and  only 


the  former  will  be  transmitted.  That  is  to  say,  there  will 
emerge  two  rays  advancing  in  the  same  line  and  with  parallel 
vibrations  ;  hence  they  are  capable  of  interfering,  since  light- 
rays  can  completely  interfere  only  when  their  vibration  are  in 
the  same  plane.  If  these  vibrations  are  in  the  same  phase,  the 
intensity  of  the  resultant  wave  will  be  proportional  to  the 
square  of  the  sum  of  their  amplitudes  ;  but  if  in  opposite 
phases,  the  intensity  will  be  proportional  to  the  square  of  the 
difference. 


WITH  PARALLEL  MONOCHROMATIC  LIGHT  AND 
CROSSED  NICOLS. 

In  sections  cut  perpendicular  to  the  optic  axis,  interference 
phenomena  are  impossible,  and  the  field  remains  dark  through- 
out a  rotation  of  360°,  since  the  light  from  the  polarizer 
traverses  the  section  in  the  direction  of  the  optic  axis,  and 
therefore  without  change.  In  all  other  sections  there  is  double 
refraction,  and  consequently  interference.  The  field  is  dark 
four  times  during  the  entire  rotation  of  the  stage  at  intervals 
of  90° ;  that  is,  when  the  planes  of  vibration  of  the  rays  pro- 
duced in  the  section  coincide  with  the  principal  sections  of 
either  nicol.  For  all  other  positions  the  field  is  illuminated  by 
the  components  of  the  rays  which  penetrate  the  analyzer,  and 
this  brightening  is  most  intense  in  the  diagonal  positions. 

The  rays  pursuing  the  same  path  are  brought  into  one  plane 
of  vibration  by  the  analyzer  and  there  interfere.  The  kind  of 
interference  is  determined  by  the  formula, 


in  which  A   is  the  difference  in  the  retardations  in  microns 
which  the  two  rays  have  undergone. 

h  is  the  thickness  of  the  section  in  microns. 

A,  is  the  wave-length  in  microns. 

i   is  the  angle  of  incidence. 

nj  is  the  index  of  refraction  for  the  slower  ray. 

n2  is  the  index  of  refraction  for  the  faster  ray. 

For  normal  incidence,  which  is  usually  the  case  in  practice, 
A  =  h  (HI  —  n2). 

7 


FIG.  2. 


When  A  =  A., 
2  A,  3  A.,  etc., 
the  field  is  dark 
during  an  entire 
revolution  of 
the  section  on 
the  stage.  From 
Fig.  2,  it  will  be 
seen  that  the 
A  components  o  f 
PPi ,  on  emer- 
gence from  the 
plate  with  vibra- 
tion directions 
OOx  and  EE1} 
must  be  of  the 
same  phase; 
that  is,  the  sim- 
ultaneously dis- 
placing forces 


acting  upon  any  ether  particle  C  are  Co  and  Ce,  which  when 
reduced  to  the  plane  of  the  analyzer  are  Ca  and  Cai,  equal 
and  in  opposite  directions. 

When  A  =1  A., 
•f  A,  f  A.,  etc. ,  the 
light  will  be  at 
its  brightest  be- 
cause the  com- 
ponents of  PPi 
must  then  be  of 
opposite  phase 
A  on  emergence 
from  the  plate. 
From  Fig.  3,  the 
simultaneous- 
ly displacing 
forces  acting  on 
any  ether  parti- 
cle C  are  Co  and 
Ce,  which  re- 
duced by  the 


analyzer  to  its  plane  are  Ca  and  Ca1?  equal  and  in  the  same 
direction. 

Let  a  thin  wedge  of  doubly-refracting  crystal,  such  as  quartz, 
be  cut  so  that  its  planes  of  vibration  are  parallel  to  the  length 
and  breadth,  and  then  examined  between  crossed  nicols,  using 
perpendicularly  incident  monochromatic  light.  Upon  rotation, 
it  will  appear  dark  when  in  the  "normal  positions,"  but  in  all 
others  it  will  show  a  series  of  dark  and  light  bands  which  are 
most  marked  in  the  diagonal  position.  When  the  nicols  are 
made  parallel  the  portions  formerly  light  become  dark,  and 
vice  versa.  The  distance  between  the  dark  bands  varies  with 
light  of  different  wave-lengths. 

The  formula  for  intensity  of  the  emerging  light  is 

I  =  a2  sin2  2  IF  sin2 


in  which  a  —  the  amplitude  of  the  incident  wave,  A.  =  the  wave- 
length, A  =  the  retardation,  V  =  the  angle  between  the  vibra- 
tion plane  of  the  polarizer  and  the  slower  wave. 

I  will  be  a  minimum  : 

(a)  when  sin2  2  V  =  0,  or  2  W  =  0°,  180°,  360°,  540°,  etc.,  or 
W  =  0°,  90°,  180°,  270°,  etc.  ;  that  is,  four  times  in  a  revolution 
of  the  section,  or  whenever  the  planes  of  vibration  of  the  plate 
coincide  with  those  of  the  polarizer  and  the  analyzer. 

(£)  when  sin3  ^—  =  0,  which  will  be  whenever  y  =  1, 
2,  3,  etc.  ;  that  is,  whenever  the  phase  difference  A  is  a  multiple 
of  A,  for  then  sin2  -^—  becomes  sin2 180°,  or  a  mutiple  there- 

l      A     J 

of.  This  is  independent  of  W ;  hence  with  this  condition,  the 
section  will  appear  dark  throughout  an  entire  revolution. 

I  will  be  a  maximum : 

When  sin2  2^  =  1  or  2W  =  90°,  270°,  etc.,  and  W  =  45°, 
135°,  225°,  etc. 

When  y  =  |,  f,  |,  etc.,  for  then  sin2  {^-]  =  sin2  90°,  sin2 
270°,  etc.,  =1. 

*  See  Derivation  of  Formulae,  page  21. 


WITH   PARALLEL  WHITE  LIGHT  AND   CROSSED 

.NICOLS. 

In  sections  cut  normal  to  the  optic  axis,  the  field  appears 
dark  throughout  a  rotation  of  360°,  the  optic  axis  being  the 
same  for  all  colors.  In  all  other  sections  there  is  extinction 
every  90°,  and  greatest  brightness  in  the  diagonal  positions ; 
but,  since  A  may  be  at  the  same  time  approximately  an  even 
multiple  of  i  A,  and  an  odd  multiple  of  £  A.1,  light  of  one  wave- 
length may  be  greatly  weakened  while  that  of  another  wave- 
length is  practically  undimmed  ;  that  is,  there  will  result  a  tint 
due  to  the  partial  extinction  of  some  colors. 

For  example,  suppose  that  in  a  polarization  microscope, 
parallel  white  light  passes  upward  through  the  polarizer,  whose 
principal  section  is  represented  by  PP^  in  Fig.  2.  Next  let  this 
light,  which  is  polarized  in  a  single  plane  PPi ,  pass  through  a 
thin  section  of  gypsum.  It  will  there  be  separated  into  two 
waves  vibrating  in  planes  at  right  angles  to  each  other,  OOi 
and  EEj.  The  two  waves  travel  through  the  section  with 
unequal  velocity,  and  on  emerging  one  is  retarded  a  certain 
number  of  wave-lengths,  or  fractions  thereof,  as  compared  with 
the  other.  Now  let  these  light  waves  pass  through  the  analyzer, 
with  its  vibration-plane  AAi  at  right  angles  to  PPi.  Then 
each  of  the  two  sets  of  vibrations  will  have  a  component  in  the 
direction  AAi,  and  these  will  emerge  polarized  in  the  same 
plane,  and  are  therefore  capable  of  interfering.  Light  corre- 
sponding to  the  other  components  in  the  direction  PPX  will  be 
extinguished.  As  previously  stated,  one  of  the  emergent  waves 
is  slightly  retarded  as  compared  with  the  other.  The  amount 
of  this  retardation,  on  which  the  interference-color  of  the  sec- 
tion depends,  is  proportional  to  the  difference  of  the  indices, 
and  also  to  the  thickness  of  the  section. 

If  in  the  examination  of  the  plate  of  gypsum  above,  one  of 
the  nicols  had  been  rotated  90°, — that  is  to  say,  if  the  principal 
sections  of  the  two  nicols  had  been  parallel, — interference  would 
have  taken  place  between  the  emerging  rays  ;  but  the  color 
resulting  in  each  case  would  have  been  exactly  complementary 
to  that  obtained  at  first  when  the  nicols  were  crossed.  For 
instance,  in  Fig  2,  if  W  =  o,  the  principal  sections  of  the  plate 
coincide  with  those  of  polarizer  and  analyzer,  and  complete  ex- 
tinction occurs  ;  but  if  A AI  be  made  parallel  to  PPi ,  evidently 
the  section  would  appear  white. 

10 


If  we  examine  in  the  same  way  a  very  thin  and  gradually- 
tapering  wedge  of  some  doubly-refracting  crystal  (quartz  for 
example),  cut  so  that  its  planes  of  vibration  are  parallel  to  the 
length  and  breadth,  the  successive  interference-colors  of  the  first 
order,  beginning  at  the  thin  end,  pass  from  an  iron-gray  through 
bluish-gray  to  white,  yellow,  and  red ;  then  follow  violet, 
indigo,  blue,  green,  yellow,  orange,  and  red  of  the  second  order; 
then  the  similar  but  paler  series  of  colors  of  the  third  order, 
and  finally  the  very  pale  shades  of  green  and  red  of  the  fourth 
order.  Beyond  this  the  colors  are  not  very  distinct,  and  white 
of  a  higher  order  finally  results  from  the  interference.  A  min- 
eraj  of  very  strong  double  refraction,  such  as  calcite,  shows 
only  the  white  of  the  higher  orders  unless  extremely  thin. 

In  general,  a  thin  section  of  a  doubly -refracting  crystal,  ex- 
amined between  crossed  nicols,  is  not  dark  except  when  placed 
in  certain  definite  positions.  In  any  other  position  it  does  not 
completely  extinguish  the  light,  but  its  effect,  in  conjunction 
with  the  nicols,  is  partially  to  suppress  the  several  components 
of  the  white  light  in  different  degrees,  so  that  in  the  emergent 
beam  these  components  are  no  longer  in  the  proportions  to  give 
white  light.  In  this  way  arise  interference-tints,  which  may  be 
definitely  classed  according  to  Newton's  color-scale.  (See  page 
15.)  The  several  tints,  though  graduating  into  one  another, 
are  distinguished  by  names  and  divided  into  several  orders. 
The  precise  position  in  the  scale  of  a  given  tint  observed  be- 
tween crossed  nicols  can  be  fixed  by  means  of  a  quartz-wedge 
or  other  contrivance  for  compensating  or  neutralizing  the  effect 
of  double  refraction  of  the  section. 

The  interference-tints  given  by  a  crystal  section  depend 
(1)  on  the  amount  of  double  refraction  of  the  mineral  (nj  —  n2), 
which  is  a  specific  character  ;  (2)  on  the  direction  of  the  section 
relatively  to  the  ellipsoid  of  optic  elasticity,  the  tint  being 
highest  for  a  section  parallel  to  the  greatest  and  least  axes  of 
the  ellipsoid  ;  (3)  on  the  thickness  of  the  section.  The  last  two 
are  disturbing  factors,  which  must  be  eliminated  before  we  can 
use  the  interference-tints  as  an  index  of  the  amount  of  double 
refraction  of  the  crystal,  and  as  a  useful  criterion  in  identifying 
the  mineral.  The  fact  that  the  interference-tints  depend  in 
part  on  the  direction  of  the  section  through  the  crystal  will 
give  little  difficulty  in  estimating  approximately  the  amount  of 
double  refraction  of  the  mineral.  If  several  crystals  of  the 

11 


same  mineral  are  contained  in  a  rock-section,  it  is  sufficient  to 
have  regard  to  the  ones  which  give  the  highest  interference- 
tints.  Even  a  single  crystal  will  in  the  majority  of  cases  give 
tints  not  so  far  below  those  proper  to  the  mineral  as  to  occasion 
error,  but  the  possibility  of  the  section  having  an  unfavorable 
direction  must  be  borne  in  mind. 

In  making  a  determination,  the  manner  of  proceedure  is 
briefly  as  follows :  Select  a  crystal,  revolve  it  on  the  stage  of 
the  microscope  until  it  is  in  the  position  of  maximum  intensity 
(i.  e.,  when  the  greatest  and  least  axes  of  elasticity  make  an 
angle  of  about  45°  with  the  principal  sections  of  polarizer  and 
analyzer)  ;  introduce  the  quartz-wedge  between  polarizer  and 
analyzer  (in  slot  provided  for  the  purpose,  also  making  an  angle 
of  45°  with  principal  sections  of  the  two  nicols),  until  a  position 
of  maximum  darkness  of  the  crystal  is  reached.  If  the  inter- 
ference-tint of  the  crystal  cannot  in  this  position,  or  when 
revolved  through  an  angle  of  90°,  be  reduced  through  the  suc- 
cessive colors  of  the  scale,  in  descending  order,  back  to  dark- 
ness, the  section  is  an  unfavorable  one,  i.  e. ,  it  is  not  cut  ap- 
proxmiately  parallel  to  the  plane  of  the  axes  of  greatest  and 
least  elasticity  of  the  crystal.  When  a  favorable  section  is 
found,  after  having  fixed  the  quartz-wedge  in  the  position 
giving  maximum  darkness  to  the  crystal,  first  remove  the  sec- 
tion from  the  stage  of  the  microscope.  Now  the  quartz- wedge, 
as  viewed  through  the  microscope,  will  be  of  the  same  color 
and  order  in  Newton's  scale  as  that  of  the  crystal-section  under 
the  same  conditions.  Carefully  withdraw  the  wedge,  at  the 
same  time  noting  the  succession  of  tints  in  descending  order.  In 
this  fashion,  one  can  estimate  the  order  of  the  interference-color 
in  the  scale  ;  and  by  applying  the  formula  A  =  h  (nx  —  n2), 
in  connection  with  the  tables  given  on  pages  15  and  16,  many 
of  the  commoner  rock-forming  minerals  may  be  identified. 

The  following  twenty-four  examples  will  illustrate : 


12 


Rock  Section. 

Thick- 
ness 
in  Mi- 
crons. 

Crystals. 

Interference  Color 
between 
Crossed  Nicols. 

Interference  Color 
between 
Parallel  Nicols. 

Order 
of 
Color. 

QUARTZ-  PORPHYBY  .  .  . 
RHYOLITE            .  . 

26.5 
32  8 

Quartz 
Orthoclase 

Greenish  white 
Gray 

Straw  -  yellow 

Brown 
Brownish  yellow 

I 
I 

23  6 

Orthoclase 
Biotite 
Hornblende 

Almost  pure  white 
Light  greenish  gray 
Light  green 

Gray 

Light  red 
Grayish  red 
Carmine  -red 

I 
IV 
II 

I 

PEGMATITE  

21  4 

Orthoclase 
Plagioclase 
Microcline 
Biotite 
Muscovite 
Hornblende 
Augite 
Hypersthene 
Tourmaline 
Zircon 
Apatite 

Bluish  gray 
Bluish  gray 
Bluish  gray 
Greenish  yellow 
Bright  orange-  red 
Purple 
Reddish  orange 
Pale  straw  -yellow 
Brownish  yellow 
Greenish  yellow 
Lavender-gray 

Brownish  white 
Brownish  white 
Brownish  white 
Grayish  blue 
Greenish  blue 
Light  green 
Bluish  green 
Dark  reddish  brown 
Gray-blue 
Grayish  blue 
Yellowish  white 

I 
I 
I 
III 
II 
II 
I 
I 
I 
III 
I 

I' 

TRACHYTE  •       

27  2 

Feldspar 
Biotite 
Muscovite 
Tourmaline 
Beryl 

Grayish  blue 
Greenish  blue 
Pure  yellow 
Brownish  yellow 
Grayish  blue 

Brownish  white 
Flesh-colored 
Indigo 
Gray-blue 
Brownish  white 

I 
III 
II 

I 
I 

PHONOLiITE  

33  1 

Biotite 
Hornblende 
Augite 

Dull  purple 
Sky-blue 
Indigo 

Dull  sea-green 
Orange 
Golden  yellow 

III 
II 
II 

TlNGUAITE  

18  4 

Augite 
Elaeolite 

Greenish  blue 
Grayish  blue 

Brownish  orange 
Brownish  white 

II 
I 

SYENITE                  .  . 

24  7 

Biotite 
Hornblende 
Augite 
Leucite 
Elaeolite 

Orthoclase 

Dark  violet-red 
Bright  yellow 
Brownish  yellow 
Black 
La  vender  -gray 

Bluish  gray 

Green 
Blue 
Gray-blue 
Bright  white 
Yellowish  white 

Brownish  yellow 

"ll 

I 
I 
I 
I 

I 

NEPHELINE-SYENITE  . 
ANDESITE    . 

19.6 
25  3 

Augite 
Hornblende 
Biotite 

Orthoclase 
Nepheline 
Hornblende 
Augite 
Biotite 

Red 
Indigo 
Flesh  -colored 

Grayish  blue 
Lavender-gray 
Reddish  orange 
Brownish  yellow 
Indigo 

Pale  green 
Golden  yellow 
Sea-green 

Brownish  white 
Yellowish  white 
Bluish  green 
Gray-blue 
Impure  yellow 

I 
II 
III 

I 
I 
I 
I 
III 

I 

Augite 
Biotite 
Hornblende 

Deep  red 
Carmine-red 
Indigo 

Yellowish  green 
Green 
Golden  yellow 

I 
III 
II 

13 


Rock  Section. 

Thick- 
ness 
in  Mi- 
crons. 

Crystals. 

Interference  Color 
between 
Crossed  Nicols. 

Interference  Color 
between 
Parallel  Nicols. 

Order 
of 
Color. 

DACITE  

33  7 

Quartz 

Light  yellow 

Indigo 

I 

DIOBITE  

29.1 

Plagioclase 
Biotite 
Augite 
Hornblende 

Plagioclase 

Yellowish  white 
Whitish  gray 
Green 
Light  green 

Greenish  white 

Carmine-red 
Bluish  gray 
Light  carmine-red 
Purplish  red 

Brown 

I 
IV 
II 
II 

I 

BASALT 

38  9 

Augite 
Biotite 
Hornblende 

Plagioclase 

Sky-blue 
Bluish  green 
Greenish  blue 

Orange 
Lilac 
Brownish  orange 

Indigo 

JII 
IV 

II 

I 

OLIVINE-  BASALT  

41.7 

Augite 
Plagioclase 

Greenish  yellow 
Bright  yellow 

Violet 
Blue 

II 
I 

HORNBLENDE-BASALT  . 

NEPHELINE-BASALT  .  .  . 
LEUCITE-BASALT  

46.3 

43.6 
45  9 

Augite 
Olivine 

Plagioclase 
Augite 
Hornblende 

Nepheline 

Augite 

Leucite 

Pure  yellow 
Flesh  colored 

Bright  yellow 
Bright  orange-red 
Dark  violet  red 

Gray 
Orange 

Indigo 
Sea-green 

Blue 
Greenish  blue 
Green 

Brownish  yellow 
Dark  blue 

White 

II 
III 

I 
II 
II 

I 
II 

I 

MELILITE-BASALT  
DIABASE  

51.3 
49  2 

Augite 

Melilite 
Augite 

Bright  orange-red 

Straw-yellow 
Light  bluish  violet 

Greenish  blue 

Deep  violet 
Yellowish  green 

Gray-blue 

II 

I 
III 

I 

GABBBO 

46  5 

Augite 

Dark  violet  red 

Green 
Blue 

II 
I 

NOBITE  

41'3 

Augite 

Bright  orange-red 

Greenish  blue 
Blue 

II 
I 

TBOCTOLITE 

55  4 

Hypersthene 

Reddish  orange 

Bluish  green 
Gray-  blue 

I 

I 

THEBALITE  

33.4 

Olivine 
Augite 

Whitish  gray 

Bluish  gray 
Brownish  orange 

IV 

PEBIDOTITE  

29.8 

Biotite 
Plagioclase 
Nepheline 
Apatite 

Olivine 

Whitish  gray 
Yellowish  white 
Grayish  blue 
Grayish  blue 

Dark  violet-red 

Bluish  gray 
Carmine-red 
Brownish  white 
Brownish  white 

Green 

II 

Augite 
Diopside 
Hornblende 
Biotite 

Sky-blue 
Bright  orange-red 
Greenish  blue 
Light  green 

Orange 
Greenish  blue 
Brownish  orange 
Carmine 

II 
II 
II 

IV 

14 


COLOR-SCALE  ACCORDING  TO  NEWTON. 


A   IN 
MICRONS. 

INTERFERENCE  COLOR 

BETWEEN 

CROSSED  NICOLS. 

INTERFERENCE  COLOR 

BETWEEN 

PARALLEL  NICOLS. 

0.000 

Black 

Bright  white                  > 

0.040 

Iron-gray 

White 

0.097 

Lavender-gray 

Yellowish  white 

0.158 

Grayish  blue 

Brownish  white 

0.218 

Clearer  gray 

Brownish  yellow 

0.234 

Greenish  white 

Brown 

0.259 

Almost  pure  white 

Light  red 

a 

M 

H 

0.267 

Yellowish  white 

Carmine-red 

3 

0.275 

Pale  straw-yellow 

Dark  reddish  brown 

o 

9 

0.281 

Straw-yellow 

Deep  violet 

U 
• 

0.306 

Light  yellow 

Indigo 

a 

0.332 

Bright  yellow 

Blue 

0.430 

Brownish  yellow 

Gray-blue 

0.505 

Reddish  orange 

Bluish  green 

0.536 

Red 

Pale  green 

0.551 

Deep  red 

Yellowish  green 

0.565 

Purple 

Lighter  green 

0.575 

Violet 

Greenish  yellow 

0.589 

Indigo 

Golden  yellow 

0.664 

Sky-blue 

Orange 

0.728 

Greenish  blue 

Brownish  orange 

G0 

i 

0.747 

Green 

Light  carmine-red 

o 

a 

0.826 

Lighter  green 

Purplish  red 

.     b 

0.843 

Yellowish  green 

Violet-purple 

s 

0.866 

Greenish  yellow 

Violet 

1 

0.910 

Pure  yellow 

Indigo 

0.948 

Orange 

Dark  blue 

0.998 

Bright  orange-red 

Greenish  blue 

1.101 

Dark  violet-red 

Green 

1.128 

Light  bluish  violet 

Yellowish  green            ] 

1.151 

Indigo 

Impure  yellow 

1.258 

Greenish  blue 

Flesh-colored 

H 

1.334 

Sea-green 

Brownish  red 

H 

M 

g 

1.376 

Brilliant  green 

Violet 

U 

o 

1.426 

Greenish  yellow 

Grayish  blue 

a 

a 

1.495 

Flesh-colored 

Sea-green 

H 
33 

1.534 

Carmine-red 

Green 

1.621 

Dull  purple 

Dull  sea-green 

1.652 

Violet-gray 

Yellowish  green            -> 

1.682 

Grayish  blue 

Greenish  yellow                  o 

1.711 

Dull  sea-green 

Yellowish  gray 

1.744 

Bluish  green 

Lilac                                 -    « 

1.811 

Light  green 

Carmine 

1.927 

Light  greenish  gray 

Grayish  red 

^ 

2.007 

Whitish  gray 

Bluish  gray 

15 


DOUBLY-REFRACTING  POWER  OF  VARIOUS  MINERALS. 


CRYSTAL. 

D!—  D2 

CRYSTAL. 

nx  —  n2 

Proustite    .     .     . 

.     0.2953 

Phenacite  .     . 

.      .      0.0157 

Sulphur      .     .     . 

.     0.2900 

Anorthite  .     . 

.     .      0.0130 

Rutile    .... 

0.2871 

Natrolite    . 

.     .     0.0119 

Dolomite    .     .     . 

.     0.1791 

Hyphersthene 

.     .     0.0115 

Calcite 

0.1722 

Andalusite 

.     .     0.0110 

Brookite     .     .     . 

.     0.1580 

Gypsum     .     . 

.     .     0.0100 

Aragonite  .     .     . 

.     0.1558 

Bronzite     .     . 

.     .     0.0100 

Cassiterite      .     . 

.     0.0968 

Corundum 

.     .     0.0092 

Zircon 

0  0618 

Quartz  . 

0  0091 

Biotite  .... 

0.0600 

Topaz     . 

.     .     0  0090 

Epidote      .     .     . 

.     0.0545 

Cordierite  .     . 

.     .     0.0090 

Anhydrite       .     . 

.     0.0430 

Axinite 

.     .     0.0090 

Muscovite       .     . 

.     0.0420 

Orthoclase 

.     .     0.0080 

Olivine 

0.0360 

Albite    .     .     . 

0.0080 

Scapolite 

.     0.0360 

Beryl     .     . 

0.0073 

Diopside    .     .     . 

.     0.0335 

Plagioclase 

.     .     0.0071 

Cancrinite  . 

.     0.0289 

Zoisite  . 

0.0060 

Tremolite  .     .     . 

.     0.0265 

Melilite      .     . 

.     .     0.0058 

Hornblende     .     . 

.     0.0240 

Nephelite  .     . 

.     .     0.0049 

Augite  .... 

.     0.0220 

Elaeolite    .     . 

.     .     0.0047 

Sillimanite      .     . 

.     0.0220 

Apatite       .     . 

.     .     0.0042 

Glaucophane  .     . 

.     0.0216 

Vesuvianite     . 

.     .     0.0020 

Brucite       .     .     . 

.     0.0210 

Apophyllite    . 

.     .     0.0019 

Tourmaline     .     . 

.     0.0189 

Leucite       .     . 

.     .     0.0010 

Chlorite     .     . 

.     .     0.0010 

16 


Plate  2. 


PLATE  2.     (Crossed  Nicols.) 

Chalcedony, 

Section  cut  at  right  angles  to  the  length  of  a  stalactite  ;  X  100  diameters. 
Central  sections  through  the  concretions  give  a  spherulitic  interference 
cross.  Radial  fibers  appear  to  be  uniaxial,  since  extinction  is  complete 
in  fibers  parallel  to  planes  of  vibration  of  polarizer  and  analyzer,  i.  e. , 
when  y  —  o  ;  the  illumination  is  at  a  maximum  when  W  =  45°. 
Extinction  is  also  complete  where  fibers  are  cut  at  right  angles,  for  the 
light  travels  in  the  direction  of  their  optic  axes.  The  crystal  system  of 
this  mineral  has  not  yet  been  definitely  determined ;  however,  it  appears 
to  be  optically  uniaxial. 


Plate  II. 


\ 


DKRR 


DERIVATION  OF  FORMULAE. 

Let  there  be  a  Y 
crystalline  plate 
having  OX  and 
OY  for  axes  of 
elasticity,  OP  the 
direction  of  a  pol- 
arizer on  this  sec- 
tion (fig.  5)  ;  a  ray 
of  light  polarized 
by  OP  vibrating  in 
the  principal  sec- 
tion of  the  polari- 
zer is  divided,  in 
the  plate,  into  two 
rays,  which  may 
be  regarded  as 
superposed  and  l 
which  have  at  emergence  a  difference  of  phase  : 

9  =  ^  -  (ne  —  n0). 
A-  cos  r 

Let  us  call  x  and  y  the  vibrations  of  these  rays  which  give  as 
a  resultant  an  elliptical  movement,  and  let  us  receive  the  corres- 
ponding doubly-refracted  ray  on  an  analyzer  having  OA  for 
the  direction  on  the  plane  of  the  plate  ;  the  angle  between 
the  polarizer  and  the  axis  OX  of  the  plate  is  POX.  Put 
Z  POX  =  ft.  The  angle  between  the  analyzer  and  the  same 
axis  is  AOX,  and  we  put  Z  AOX  =  a.  The  components  be- 
longing to  OX  and  OY  are  : 


x  =  a  cos  2 


cos 


y  =  a  cos  2  n  _  .  sin  ft. 

They  give  rise  to  two  rays  which  emerge  from  the  plate  with 
a  difference  of  phase  <p  ;  the  components  of  these  rays  are  : 

x  =  a  cos  2  n  _  cos  /?,  y  =  a  cos  2  n    —  —  <p    sin  ft. 

x  gives  to  OA  a  component  £  =  x  cos  a  parallel  to  the  prin- 
cipal section  and  a  perpendicular  ;  y  gives  to  OA  a  component 
77  =  y  sin  a  parallel  to  its  principal  section  and  a  perpendicular 
to  that  section.  We  need  not  concern  ourselves  with  the  per- 
pendiculars, which  are  eliminated;  the  parallels  give  a  result- 
ant, which  is  their  algebraic  sum  : 

£  +  rf  =  x  cos  a  -f-  y  sin  a. 
19 


Replacing  x  and  y  by  their  values,  we  have 

%  -f  rj  =  a  cos  a  cos  ft  cos  2  n  —  -f  a  sin  a  sin  yS  cos  2^1-—-  —  q>  \ 


=  a  cos  2  TT  —  r  (cos  a  cos  ft  -f-  sin  ar  sin  ft  cos  %  ft  q>) 

-f-  a  sin  2  TT  —  sin  a  sin  /?  sin  2  n  (p. 

According  to  Fresnel,  let  us  put 

a  (cos  ot  cos  ft  -\-  sin  a  sin  ft  cos  2  n  qt)  =  E  cos  A? 
a  sin  <*  sin  ft  sin  2  ?r  <p  =  E  sin  A  . 

rj  +  £  =  cos  2  TT  —  cos  A  -{-  sin  2  TT  A  sin  A  ==  cos    2  n  A  —  A    . 

This  relation  shows  that  the  amplitude  of  vibration  is  E,  if 
we  compare  it  to  x  =  a  cos  2  n  —  . 

We  have  : 

™  (    ^  (COS  a  COS  ^  +  Sil 

cos2  A  +  sm2  A    =  E«=  ft  Si 


E2 

==  cos2  «  cos2  /?  -f-  sin2  a  sin2  /?  cos2  2  JT  ^ 
a2 

-f  2  cos  «  cos  ft  sin  #  sin  ^  cos  2  TT  <p  -\-  sin2  <*  sin2  ft  sin2  2  TT  ^> 
=  cos2  a  cos2  /?  -f  sin2  a  sin2  y#  -f-  2  sin  «  sin  ft  cos  <*  cos  /? 

-  4  sin  or  sin  ft  cos  or  cos  /?  sin2  rt  cpt 
Replacing  cos  2  n  cp  by  1  —  2  sin2  TT  9?, 

E2 

—  —  (cos  «  cos  ft  -f-  sin  «  sin  ft)*—  sin  2  «  sin  2  /?  sin2  TT  q> 
a 

=  cos2  («  —  ft]  —  sin  2  a  sin  2  /?  sin2  n  cp   .....     (1) 


DISCUSSION  OF  THE  FORMULA. 

We  have  that  q>  =  —- (ne  —  n0). 

A  cos  r 

h  represents  the  thickness  of  the  section ;  r  the  angle  between 
the  rays,  which  we  regard  as  superposed,  and  the  normal  to  the 
faces  of  the  section  ;  ne ,  n0  the  indices  of  refraction  of  the 
two  rays. 

When  white  light,  composed  of  an  infinite  number  of  lights 
of  very  different  wave-lengths,  is  employed,  the  second  term 
corresponds  to  the  sum  of  the  intensities  of  these  lights,  and 
the  general  formula  may  then  be  written  : 

^!  =  cos2  (a— ft}—  sin  2  a  sin  2  ft  Ssin2  n  — ^—  (ne  —  n0)  .  -  (2) 
a2  *-  A.  cos  r 

20 


THE  DIFFERENT  CASES. 

« 

I.     a  —  ft  =  r,  case  where  the  principal  sections  of  the  po- 

2i 

larizer  and  analyzer  are  at  right  angles  to  each  other. 
Since  ft  =  a  -  —  ,  formula  (2)  becomes  : 

f*  =  cos2  -^    -  sin  2  a  sin  (2  a  —  TT)  ^>  sin2  n  cp,    or 

a 

PARTICULAR  CASES. 


=  sin2  2  a  5  sin2  TT  <p  =  sin2  2  «  S  sin2  f    n  A    1  .    (3) 

2  L  A  cos  r  J 


1°.     Let  p  =  o,  then  a  =  *.     The  axes  OX,  OY  of  the  plate 

2 

are  parallel  to  the  principal  sections  of  both  analyzer  and  polar- 
izer. All  the  colors  are  extinguished  in  the  crystal  at  the  same 
time,  since  they  have  the  same  axes  of  elasticity. 

E2 


2°.     ft  =  45°,  a  =  ~  +  45°. 
2 

^  =  ^sm27T<7> (4) 

a 

II.     a  =  /?,  the  polarizer  and  analyzer  are  parallel. 

TT2 

—2  =  1  —  sin2  2  a  ^>  sin2  ?r  <p (5) 

The  values  (3)  and  (5)  are  complementary. 

2°.     a  =  45°,  ft  =  45°. 

—  =  1  —  ^  sin2  TT  <p (6) 

a2 

All  the  colors  have  their  maximum  of  intensity. 


If  we  employ  a  rhombohedron  of  calcite  instead  of  a  nicol, 

we  have  two  images.     The  intensity  -^  of  the  ordinary  image 

ar 

is  obtained  by  means  of  formula  (2)  by  replacing  a  by  a  -f  —  ; 

2 
the  formula  becomes : 

-  sin2  a  cos2  ft  +  sin2  ft  cos2  a 
a 

-  2  sin  a  sin  ft  cos  a  cos  ft  cos  2  TC  cp 

=  sin2  (a  —  ft)  -f  sin  2  or  sin  2  /?  ^  sin2  n  (p.    .    .    (T) 

These  phenomena  are  complementary  to  those  which  give  =^. 

a 
21 


BIBLIOGRAPHY. 

"  Theory  of  Light," PRESTON. 

u  Lehrbuch  der  Optik. " DRUDE. 

"  Light," TAIT. 

"  Physical  Optics," GLAZEBROOK. 

"  General  Physics," HASTINGS  and  BEACH. 

"  Mikroskopische  Physiographie  der  pet- 

rographisch  wichtigen  Mineralien,   .  ROSENBUSCH. 


22 


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